While the traditional geometric perspective of dynamical systems describes the topological organization of trajectories of the systems, the Koopman operator gives another perspective of dynamical systems on the evolution of observables. This perspective was originally introduced by Koopman [1]} and later extended to dissipative systems by Mezic [2]}. We here formally discuss a generic way of deriving the transformation of \(X_i(t)\)  to \(\theta _i(t)\) , frequency \(\omega \) , and PCF \(\Gamma _i\)  in Eq. (REF ) using the Koopman operator in a vector-valued function space.
